Steven Naismith struck the winner as Hearts ended Hibernian's hopes of finishing second in the Ladbrokes Premiership.
A 2-1 defeat at Tynecastle means Hibs will finish fourth unless they beat Rangers by six goals at Easter Road on Sunday.
Naismith headed home in the 57th minute after Florian Kamberi's penalty had cancelled out Kyle Lafferty's opening goal.
Hibs only needed a point to keep in touch with second-placed Aberdeen but they rarely threatened a second equaliser.
They are now relying on Celtic to beat Motherwell in the William Hill Scottish Cup final to claim a Europa League spot, unless they pull off an extraordinary result against third-placed Rangers.
Hibs made the brighter start with Brandon Barker looking the player likeliest to make a breakthrough. The winger had one shot blocked after a piercing run and then forced a decent stop from Jon McLaughlin.
The simmering derby tension boiled over when Lafferty went in hard and late on Lewis Stevenson, sparking a bout of pushing and a yellow card for the Northern Ireland international.
Hearts channelled their aggression well and went ahead in the 26th minute when Lafferty got the better of Stevenson to get on the end of Naismith's through ball. The striker poked the ball beyond Ofir Marciano into the corner of the net from 10 yards.
Lafferty celebrated his 19th goal of the season in front of the visiting fans and signalled he was unconcerned by the ensuing missiles.
The action was unrelenting before the break. Harry Cochrane, Hearts' 17-year-old midfielder, was more than holding his own amid the rancour and he showed skill and drive to twice get into shooting positions. Marciano saved his first effort and a second deflected just wide.
Barker and Scott Allan forced saves at the other end but Hibs suffered a blow before the interval when wide player Martin Boyle went off with a hamstring injury. Dylan McGeouch came on.
Hibs made the perfect start to the second half as Stevenson atoned for his part in the opener by robbing both Michael Smith and Joaquim Adao before being tripped by the latter. Kamberi sent McLaughlin the wrong way from the 48th-minute spot-kick.
But the hosts went back in front when Naismith glanced a header home from Cochrane's free-kick.
Hearts generally held firm, although the visitors did create some chances
Barker took the ball too wide from Vykintas Slivka's pass and shot over before McLaughlin denied substitute Jamie Maclaren following a late corner, and Kamberi shot just past the top corner deep into six minutes of injury time.